Stability of differentially heated flow from a rotating sphere

Abstract

h i g h l i g h t s• Differentially heated flow of a thin fluid layer from a rotating sphere has been investigated.• A numerical solution procedure for solving the steady and unsteady equations has been proposed. • An approximate analytical solution has been derived. • A linear stability analysis has estimated a theoretical value for the onset of instability. • Good agreement was found between numerical, analytical and theoretical results. a b s t r a c tWe present results on the flow of a thin fluid layer over a rotating sphere having a surface temperature that varies in the latitudinal direction. The fluid is taken to be viscous, incompressible and Newtonian while the flow is assumed to possess both azimuthal and equatorial symmetry. The governing Navier-Stokes and energy equations are formulated in terms of a stream function and vorticity and are solved subject to no-slip boundary conditions. An approximate analytical solution for the steady-state flow has been derived and is compared with numerical solutions to the steady and limiting unsteady equations. For small Rayleigh numbers these solutions are found to be in close agreement. However, as the Rayleigh number is increased noticeable differences occur. A numerical solution procedure is presented and a linear stability analysis has been conducted to predict the onset of instability. Good agreement between the theoretical predictions and the observed numerical simulations was found.

An analytical study of the early stages of unsteady free convective flow from a differentially heated rotating sphere at large Grashof numbers

“…Free convective flow from a differentially heated rotating sphere occurs naturally in the atmosphere and thus represents an important and well-studied problem. Numerous investigations, some of which are listed in [1], focussing on various aspects have been devoted to this subject. The present study represents a continuation of the work reported by D'Alessio et al [1] which emphasized the stability of the steady-state flow for small Rayleigh numbers.…”

“…In the spherical coordinates (r, q ) and cast in dimensionless form the equations can be formulated as [1,2] In the above t denotes time, r is the radial coordinate, and q is the angle with the polar axis. The flow variable W denotes the scaled zonal velocity while T is the scaled temperature.…”

Mathematical modeling and computational methods

“…Authors show the capability of the proposed method by one-dimensional and three-dimensional benchmark problems. Serge D'Alessio et al [17] investigate differentially heated flow of a thin fluid layer from a rotating sphere. They construct a numerical solution procedure for solving the steady and unsteady equations including an approximate analytical solution of the problem.…”